A Characterization of Absolute Neighborhood Retracts

By an absolute neighborhood retract (ANR) I mean a separable metrizable space which is a neighborhood retract of every separable metrizable space which contains it and in which it is closed. This generalization of Borsuk's original definition was given by Kuratowski for the purpose of enlarging the class of absolute neighborhood retracts to include certain spaces which are not compact. The space originally designated by Borsuk as absolute neighborhood retracts (or $K-sets) will now be referred to as compact absolute neighborhood retracts. Many of the properties of compact ANR-sets hold equally for the more general ANR-sets. The Hubert parallelotope Q, that is, the product of the closed unit interval [0, 1 ] with itself a countable number of times is a "universal" compact ANR in the sense that every compact ANR is homeomorphic to a neighborhood retract of Q. The classical theory of Borsuk makes good use of the imbedding of compact ANR-sets in Q. The problem solved here is that of finding a "universal" ANR.